scholarly journals Pseudoinversion of degenerate metrics

2003 ◽  
Vol 2003 (55) ◽  
pp. 3479-3501 ◽  
Author(s):  
C. Atindogbe ◽  
J.-P. Ezin ◽  
Joël Tossa
Keyword(s):  
Differential Geometry ◽  
Large Class ◽  
Smooth Manifold ◽  
Differential Operators ◽  
Type Operator ◽  
Lorentzian Space ◽  
Lightlike Hypersurface ◽  
Definition Of ◽  
Lightlike Hypersurfaces ◽  
Theoretical Results

Let(M,g)be a smooth manifoldMendowed with a metricg. A large class of differential operators in differential geometry is intrinsically defined by means of the dual metricg∗on the dual bundleTM∗of 1-forms onM. If the metricgis (semi)-Riemannian, the metricg∗is just the inverse ofg. This paper studies the definition of the above-mentioned geometric differential operators in the case of manifolds endowed with degenerate metrics for whichg∗is not defined. We apply the theoretical results to Laplacian-type operator on a lightlike hypersurface to deduce a Takahashi-like theorem (Takahashi (1966)) for lightlike hypersurfaces in Lorentzian spaceℝ1n+2.

Some misinterpretations and lack of understanding in differential operators with no singular kernels

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 594-612 ◽  
Author(s):  
Abdon Atangana ◽  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Calculus ◽  
Power Law ◽  
Initial Value Problems ◽  
Differential Operators ◽  
Integral Operators ◽  
Initial Value ◽  
Complex Processes ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Theoretical Results

AbstractHumans are part of nature, and as nature existed before mankind, mathematics was created by humans with the main aim to analyze, understand and predict behaviors observed in nature. However, besides this aspect, mathematicians have introduced some laws helping them to obtain some theoretical results that may not have physical meaning or even a representation in nature. This is also the case in the field of fractional calculus in which the main aim was to capture more complex processes observed in nature. Some laws were imposed and some operators were misused, such as, for example, the Riemann–Liouville and Caputo derivatives that are power-law-based derivatives and have been used to model problems with no power law process. To solve this problem, new differential operators depicting different processes were introduced. This article aims to clarify some misunderstandings about the use of fractional differential and integral operators with non-singular kernels. Additionally, we suggest some numerical discretizations for the new differential operators to be used when dealing with initial value problems. Applications of some nature processes are provided.

Edges of Regression and Limit Normal Point of Conjugate Surfaces1

1998 ◽  
Vol 122 (4) ◽  
pp. 419-425 ◽  
Author(s):  
Ningxin Chen
Keyword(s):  
Differential Geometry ◽  
Contact Boundary ◽  
Boundary Line ◽  
Tangent Point ◽  
Common Tangent ◽  
Numerical Examples ◽  
Envelope Surface ◽  
The Common ◽  
Conjugate Surfaces ◽  
Definition Of

The presented paper utilizes the basic theory of the envelope surface in differential geometry to investigate the undercutting line, the contact boundary line and the limit normal point of conjugate surfaces in gearing. It is proved that (1) the edges of regression of the envelope surfaces are the undercutting line and the contact boundary line in theory of gearing respectively, and (2) the limit normal point is the common tangent point of the two edges of regression of the conjugate surfaces. New equations for the undercutting line, the contact boundary line and the limit normal point of the conjugate surfaces are developed based on the definition of the edges of regression. Numerical examples are taken for illustration of the above-mentioned concepts and equations. [S1050-0472(00)00104-5]

scholarly journals The SL(2, C) Casson Invariant for Knots and the Â-polynomial

2016 ◽  
Vol 68 (1) ◽  
pp. 3-23 ◽  
Author(s):  
Hans U. Boden ◽  
Cynthia L. Curtis
Keyword(s):  
Large Class ◽  
Product Formula ◽  
Integral Homology ◽  
Connected Sum ◽  
Connected Sums ◽  
Knot Invariant ◽  
Definition Of ◽  
Arbitrary Knots

AbstractIn this paper, we extend the definition of the SL(2,ℂ) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.

scholarly journals Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes

ISRN Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Junhong Dong ◽  
Ximin Liu
Keyword(s):  
Mean Curvature ◽  
Higher Order ◽  
Totally Umbilical ◽  
Lightlike Hypersurface ◽  
Higher Order Mean Curvature ◽  
Lightlike Hypersurfaces ◽  
The Relationship

We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle.

Scaled quantum mechanical force fields for trifluoromethyl selenium derivatives. I. The CF3SeCN and CF3SeCH3 molecules

2006 ◽  
Vol 84 (12) ◽  
pp. 1626-1631 ◽  
Author(s):  
L E Fernández ◽  
E L Varetti
Keyword(s):  
Density Functional ◽  
Force Fields ◽  
Chemical Species ◽  
Mechanical Force ◽  
Force Constants ◽  
Quantum Mechanical ◽  
Functional Theory ◽  
Definition Of ◽  
Theoretical Results ◽  
Mechanical Force Fields

Force fields and vibrational properties were calculated for the trifluoromethyl selenium derivatives, CF3SeCN and CF3SeCH3, by means of density functional theory (DFT) techniques. The existing experimental data and assignments for these molecules were confirmed by the theoretical results. These data were subsequently used in the definition of scaled quantum mechanical force fields for such chemical species. The obtained force constants are compared with results previously published for similar compounds.Key words: trifluoromethyl selenium, force constants, structure, DFT calculation.

The affine cohomology spaces and its applications

2016 ◽  
Vol 13 (02) ◽  
pp. 1650016 ◽  
Author(s):  
Nizar Ben Fraj ◽  
Ismail Laraiedh
Keyword(s):  
Large Class ◽  
Differential Operators ◽  
Lie Superalgebra ◽  
Invariant Differential Operators ◽  
Linear Differential Operators ◽  
Relative Cohomology ◽  
Linear Formula ◽  
Invariant Differential ◽  
Linear Differential ◽  
Cohomology Spaces

We compute the [Formula: see text] cohomology space of the affine Lie superalgebra [Formula: see text] on the (1,1)-dimensional real superspace with coefficient in a large class of [Formula: see text]-modules [Formula: see text]. We apply our results to the module of weight densities and the module of linear differential operators acting on a superspace of weighted densities. This work is the generalization of a result by Basdouri et al. [The linear [Formula: see text]-invariant differential operators on weighted densities on the superspace [Formula: see text] and [Formula: see text]-relative cohomology, Int. J. Geom. Meth. Mod. Phys. 10 (2013), Article ID: 1320004, 9 pp.]

scholarly journals Totally Contact Umbilical Lightlike Hypersurfaces of Indefinite -Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Rachna Rani ◽  
Rakesh Kumar ◽  
R. K. Nagaich
Keyword(s):  
Space Form ◽  
Lightlike Hypersurface ◽  
Lightlike Hypersurfaces

We study totally contact umbilical lightlike hypersurfaces of indefinite -manifolds and prove the nonexistence of totally contact umbilical lightlike hypersurface in indefinite -space form.

Eigenvalues of regular differential operators

1978 ◽  
Vol 79 (3-4) ◽  
pp. 299-305 ◽  
Author(s):  
Harold E. Benzinger
Keyword(s):  
Fine Structure ◽  
Large Class ◽  
Fourier Coefficients ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Ordinary Differential Operators

SynopsisIt is shown that the fine structure of the asymptotic estimates for the eigenvalues of a large class of ordinary differential operators, can be described in terms of the Fourier coefficients of a function of class L2.

Singular Value Decomposition Learning on Double Stiefel Manifold

2003 ◽  
Vol 13 (03) ◽  
pp. 155-170 ◽  
Author(s):  
Simone Fiori
Keyword(s):  
Differential Geometry ◽  
Singular Value Decomposition ◽  
Scientific Literature ◽  
Singular Value ◽  
Neural Computation ◽  
Stiefel Manifold ◽  
Gradient Flows ◽  
Dynamical Properties ◽  
Value Decomposition ◽  
Theoretical Results

The aim of this paper is to present a unifying view of four SVD-neural-computation techniques found in the scientific literature and to present some theoretical results on their behavior. The considered SVD neural algorithms are shown to arise as Riemannian-gradient flows on double Stiefel manifold and their geometric and dynamical properties are investigated with the help of differential geometry.

scholarly journals Objets compacts dans les topos

1986 ◽  
Vol 40 (2) ◽  
pp. 203-217 ◽  
Author(s):  
Eduardo J. Dubuc ◽  
Jacques Penon
Keyword(s):  
Differential Geometry ◽  
Topological Space ◽  
Compact Manifolds ◽  
Topological Spaces ◽  
Internal Logic ◽  
Definition Of

AbstractIt is well known that compact topological spaces are those space K for which given any point x0 in any topological space X, and a neighborhood H of the fibre -1 {x0} KXX, then there exists a neighborhood U of x0 such that -1UH. If now is an object in an arbitrary topos, in the internal logic of the topos this property means that, for any A in and B in K, we have (-1AB)=AB. We introduce this formula as a definition of compactness for objects in an arbitrary topos. Then we prove that in the gross topoi of algebraic, analytic, and differential geometry, this property characterizes exactly the complete varieties, the compact (analytic) spaces, and the compact manifolds, respectively.

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